We prove hypoellipticity in the sense of germs for the operator
\[
P
=
L
q
L
¯
q
+
L
¯
q
t
2
k
L
q
+
Q
2
,
\mathcal {P}= L_{q}\overline {L}_{q} + \overline {L}_{q}t^{2k}L_{q} +Q^{2},
\]
where
\[
L
q
=
D
t
+
i
t
q
−
1
−
Δ
x
and
Q
=
x
1
D
2
−
x
2
D
1
,
L_{q}=D_{t}+it^{q-1}\sqrt {-\Delta _{x}}\quad \text {and}\quad Q = x_{1}D_{2}-x_{2}D_{1},
\]
even though it fails to be hypoelliptic in the strong sense. The primary tool is an a priori estimate.